The well-posedness of the Cauchy problem
for a generalized nonlinear dispersive equation is studied. Local
well-posedness for data in $H^s(\mathbb R)(s>-\frac{1}{8})$ and
the global result for data in $ L^{2}(\mathbb{R})$ are obtained if
$l=2$. Moreover, for $l=3$, the problem is locally well-posed for
data in $H^s(s>\frac{1}{4}).$ The main idea is to use the Fourier
restriction norm method.